Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.     

Comparison of the boundary theory with the contact rule of phase regions and Gupta’s method

The phase boundary theory and the contact rule of phase regions are compared, and some weaknesses of the latter are manifested. The comparison between the Gupta’s method and the boundary theory method for constructing multicomponent isobaric sections is also presented.     

Observation of baker’s yeast strains used in biotransformation by atomic force microscopy

Different strains of baker’s yeast(Saccharomyces cerevisiae) were imaged with an atomic force microscope (AFM). The images of uncoated and nonfixed samples were reproducible with high-constrast and nanometer-resolution. Molecules from the polysaccharide surface of the cell wall were pictured and the distance of atoms was measured. The preparation of samples was easy, suggesting that AFM is a useful tool in this type of analyses.     

Effect of the structure of copper(II) complexes, adsorbed on the surface of SiO2, on their catalytic activity in ozone decomposition

We describe the composition, structure, and catalytic activity in the reaction of ozone decomposition for copper(II) complexes with acido ligands and immobilized Schiff’s bases (propyl benzaldimine derivatives) that are anchored on silica (silica gel, aerosil). We demonstrate methods for controlling their catalytic activity. __________ Translated from Teoreticheskaya i éksperimental’naya Khimiya, Vol. 42, No. 1, pp. 55–60, January–February, 2006.     

Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry

Ion transport problem related to controlled potential experiments in electrochemistry is studied. The problem is assumed to be superposition of diffusion and migration under the influence of an electric field. The comparative analysis are presented for three well-known models—pure diffusive (Cottrell’s), linear diffusion-migration, and nonlinear diffusion-migration (Cohn’s) models. The nonlinear model is derived by the identification problem for a nonlinear parabolic equation with nonlocal additional condition. This problem reduced to an initial-boundary value problem for nonlinear parabolic equation. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm is derived. The comparative numerical analysis for all three models shows an influence of the nonlinear migration term, the valences of oxidized and reduced oxidized species, also diffusivity to the value of the total charge. The obtained results permits one to estimate bounds of linear and nonlinear ion transport models.     

Application of Multilinear Gradient Elution for Optimization of Separation of Chlorophenols Using Derringer’s Desirability Function

Multilinear gradient elution was applied for the simultaneous optimization of resolution and analysis times of nine chlorophenols separated by HPLC. The first relationship of ln k versus φ was determined using the isocratic retention time for each analyte. For prediction of gradient retention times of analytes, the fundamental equation of gradient elution was numerically solved. Then a grid search program was used to simulate chromatograms under each new condition. Two different chromatographic goals, analysis time and minimum distance between adjacent peaks, were evaluated simultaneously using Derringer’s desirability function for each chromatogram. The sigmoid function was used to transform the optimization criteria to desirability values. Under optimal conditions, a good agreement was observed between predicted and experimental values of the chromatographic response function when analysis time was less than 40 min.     

MLC Determination of Preservatives in Cranberry Foodstuffs

Optimized conditions for the determination of 14 food preservatives were selected by using interpretative optimization strategy and Derringer’s desirability function for combining three main chromatographic goals: resolution, separation time and ruggedness. The best mobile phase for separation by micellar liquid chromatography contains 0.045 M SDS and 1.0% (v/v) 1-pentanol at pH 2.5 adjusted with trichloroacetic acid. The accuracy of the method was confirmed by analysis of spiked samples and is useful for routine analysis of food preservatives in drinks due to simple sample preparation, safety and low cost of micellar mobile phases. Presented in part by A.P. Boichenko at the 4th Black Sea Basin Conference on Analytical Chemistry, 19–23 September, 2007, Sunny Beach, Bulgaria.